ON THE INTEGRATION OF UNBOUNDED FUNCTIONS*

Transcription

1 1932.] INTEGRATION OF UNBOUNDED FUNCTIONS 123 ON THE INTEGRATION OF UNBOUNDED FUNCTIONS* BY W. M. WHYBURN 1. Introduction. The author has shown f that F. Riesz' treatment of intégration} leads in every case to the Lebesgue integral. This demonstration makes possible a complete development of the theory of Lebesgue integration from the Riesz point of view. Such a development offers a number of advantages over the usual treatment and is especially desirable when one wishes to build a Lebesgue theory on a previous treatment of the Riemann integral. The purpose of the present paper is to emphasize further the importance of the Riesz point of view in a treatment of the general subject of integration by establishing additional relations between sequences of simple functions and functions that are summable in the senses of Lebesgue, Harnack, Denjoy, Denjoy-Khintchine-Young, and Young. The terminology and notation of Riesz' paper are used. 2. Preliminary Definitions and Theorems. In this section we give a number of definitions and theorems which are well known but which are essential for the development of later theorems. Simple function.^ A function <f>{x) is said to be a simple function on X: a^xsb, if there exist n + 1 points: Xo = a<xi<x 2 < - -<x n = b, such that </>(#) =<^, a constant,onii'.xi<x<#»+i. Null set. A set of points K is said to be of measure zero if for each e>0, there exists an at most countably infinite set of intervals such that each point of K is an interior point of some one of these intervals and the sum of the lengths of the intervals * Presented to the Society, November 28, t This Bulletin, vol. 37(1931), pp Î Acta Mathematica, vol. 42(1920), pp This point of view is essentially that used in the ordinary treatment of the Riemann integral. Loc. cit. We work entirely in the real domain. If These functions have also been called horizontal or step functions. In this connection see Ettlinger, American Journal of Mathematics, vol. 48 (1926), pp

2 124 W. M. WHYBURN [February, is less than e. Such a set of points is called a null set, and a property which holds everywhere on an interval X with the exception of points of a null set is said to hold almost everywhere on X, or is said to hold on X 0. Integral of a simple function. By J l a<j>{x)dx, where <j>{x) is a simple function on X, we mean ^*»o< *Ai, where Ai = Xi+i Xi is the length of Ii. THEOREM 1. If f(x) is the limit function on X 0 of a uniformly bounded sequence [0 W (#)] of simple f unctions, then lim n ^J h a<t>n{x)dx exists, and furthermore this limit is the same for all uniformly bounded sequences of simple f unctions which approach f (x) on XQ. Theorem 1 was proved by F. Riesz.* He used the limit whose existence is asserted in this theorem as a definition of f b af(x)dx. The author has established f the following theorem which shows that Riesz' integral is identical with the Lebesgue integral. THEOREM 2. A necessary and sufficient condition that a bounded function be measurable is that it be the limit function on X 0 of a uniformly bounded sequence of simple f unctions. Riesz established J the following theorem. THEOREM 3. A necessary and sufficient condition that a bounded function f(x) be Riemann integrable on X is that there exist a sequence [</>n(#)] of simple functions which approaches f {x) uniformly almost everywhere on X, that is, uniformly in the neighborhoods of all points on XQ. Measure of a set of points. Let K be a set of points on the interval X and let f(x) = 1 at points of K and f(x) = 0 on X K. If faf(x)dx exists the point set K is said to be measurable and its measure m{k) is defined as the value of J b af{x)dx. 3. Unbounded Functions. We now consider functions which may be unbounded on their interval of definition X: a^x^b. * Loc. cit., p f W. M. Whyburn, loc. cit., pp. 561 and 564. t Loc. cit., p This is a Lebesgue integral but is defined by the Riesz method and hence uses the measure of null sets only in its definition.

3 1932.] INTEGRATION OF UNBOUNDED FUNCTIONS 125 THEOREM 4. A necessary and sufficient condition that a function f(x) be measurable on X is that there exist a sequence of simple functions that approaches f(x) on X 0. PROOF. Necessity. Let f(x) be measurable and let Fi(x) =f(x) when /(x) ^i, Fi(x)=i when f(x)>i } and Fi(x) = i when \f(x) < i. The bounded functions Fi(x), (i = l, 2, ), are measurable and hence by Theorem 2 there exist sequences [< in(#)] of simple functions which approach these functions on X 0. By EgerofFs theorem* the sequence [<t>i n (x) ] approaches Fi{x) uniformly on X except for a set of arbitrarily small measure. For each i choose an index Ni so that \(j>ini(x) Fi(x) <(l/2*) except on a set Ei of measure less than 1/2*. Let <)>i(x) = (l>in i (x) f (i = 1, 2, 3, ). We prove that [<l>i(x) ] is a sequence which approaches ƒ (x) on X 0. Let E be the subset of X on which [4>i(x) ] does not approach ƒ (x). If E is of positive measure, we can determine a number j such that the subset D of E on which /0*0 Sj is of positive measure c (otherwise E would consist of an at most countably infinite set of null sets and would therefore be of measure zero). Let j be chosen and then choose n>j such that l/2 w <c/4. We have 0 n+r (*)- F n + r (x) <l/(2 w+r ) except on a set of measure less than l/(2 n+r ), (V=l, 2, ) At each point of D, F n+r (x)=f(x), and hence with the possible exception of a subset of measure less than ^2?=il/(2 n+r ) <c/2, we have 0»+ r (*) ƒ(*) <l/(2 n+r ), 0= 1, 2, -),ond. Hence lim^c/^x) =f(x) on a subset of D of measure c/2 which contradicts the assumption that this limit does not hold at any point of D. This completes the proof of the necessity. Sufficiency. Let [0n(*O] De a sequence of simple functions which approaches ƒ(x) on X 0 and let Mi and ikf 2, Afi<ikf 2, be any two constants. Let E be the subset of X on which M\ ^f(x) ^ M 2. Let F(x) =ƒ(*) when Mi-1 ƒ(*) ^ M 2 +1, F(*) = itfi-1 when /0)<ikfi-l, JP(x) = ikf 2 +l when /(^)>M Let 6i(x) = (j>i(x) when 0*(#) lies between Mi 1 and M" 2 +l, fl<(x) = Af 1 1 when <t>i(x)^mi l 9 6i(x) = M 2 +l when 0*(a) ^M 2 +l. Now [#;(#)] is a uniformly bounded sequence of simple functions which approaches the bounded function F{x) on X 0. Hence by Theorem 2, F(x) is measurable on X and the set of points for which Mi ^ F(x) ^ M 2 is measurable. This set, how- * See Hobson, Functions o f a Real Variable, vol. 2, p. 140.

4 126 W. M, WHYBURN [February, ever, is the set E and hence E is measurable. It follows from this that ƒ (x) is measurable on X. Definition of property B. A sequence [< n(#)] of simple functions is said to have property B if for each e>0, there exists a number M such that if I(n> e) denotes the set of intervals on which \<t> n (x) \>M t then /z (w,o \(j>n(x) \dx<e, (w = l, 2, ), where fi(n,e) \<f> n (x) \dx means the sum of the integrals of 0 n (#) on the separate intervals of I(n, e). THEOREM 5. If [(j> n (x)} is a sequence of simple f unctions which has property B and which is such that lining <j> n (x) exists on X 0, then lining J h a<t> n {x)dx exists. PROOF. Let e = l. We know that \J h a(j> n {x)dx g Mi(b a) +//(n,i) \<t>n(x) \dx<mx(b a) + l for all n. The set of numbers Kn = Ja4>n(x)dx is therefore bounded. Hence there is at least one number K which either is identical with infinitely many of the numbers K n or is the limit of a subsequence of [K n ]. In either case we may pick out a sequence [6i(x) ] of simple functions from the sequence [< n (a0] in such a way that [0;(x)] approaches the same limit function/(x) as [0 n (#)L nas property J3, and is such that limi^qo di(x)dx exists. LEMMA. If [h n (x) ] and [g n (x) ] are two sequences of simple functions which approach f (x) on XQ and which have property B, then if lim nh>00 fah n (x)dx and Um n^00 J b ag n (x)dx exist, these limits are equal. PROOF OF LEMMA. Let us suppose that \im n + 0Q f b ah n (x)dx = H, lim n^o0 f b ag n (x)dx = G, and let?7 = 7e>0 be arbitrarily assigned. Choose Ni so that for all n>n% I h n (x)dx H < e, f g n (x)dx G va I \ *J a <. Let M be the larger of the two numbers M e for [Â W (#)] and kn(#)] by property B. Let J(n, e) and J'(n, e), respectively, be the sets of intervals on which \g n (x) \ >M and \h n (x)\>m. Hence I g n {x) dx <, I h n (x) I dx<, (^=1,2, ). Let G n (x) = g n (x) when \g n (x) ^ M, G n (x) = M when g n {x) > M,

5 I93 2 -] INTEGRATION OF UNBOUNDED FUNCTIONS 127 G n (x)= M when g n {x)< M\ and let H n (x)~h n (x) when \h n (x)\^m, H n (x) = M when h n (x)>m, H n (x)= M when h n (x)< M. It follows immediately that [ \G n (x) H n (x) ] is a sequence of uniformly bounded simple functions that approaches zero on X 0. By Theorems 1 and 2 we may choose an index N, N>N U such that or all n > N (4) ƒ G n{x\ H n (x) \ dx < e. We have */a t/j(n,c) g f G.(*)d* + f ^ + gn(x) ]d#,,6 Ja J H n(x)dx a I */J'(n, ) ^ f H n (x)dx + f Ja Jj'(n,c) M + h n {x) ]d#. a h n (x)dx Since If < g n (#) on /(/z, e) and M< \h n (x) on J'(n } have e), we /» b s*b,% b (2) I G n {x)dx - 2e < I gn(x)dx < I G n {x)dx + 2e, J a J a J a»&»&» b J H n (x)dx - 2e < I h n (x)dx < I H n (x)dx + 2e. a J a J a If we combine (2) and (3) and make use of (4), we obtain I gn{x)dx I h n (x)dx %) a J a + 4e < 5e, forw > iv. < I [Gn(x) - H n (x)]dx A combination of (5) with (1) yields \G-H\<7e, for all n>n. Since rj = 7e was arbitrarily chosen, it follows that G = H. PROOF OF THEOREM 5. If lim,^ J b a4> n (x)dx is not X", we may pick a subsequence [g,(x)] from [0»(#)] which approaches ƒ(x)

6 128 W. M. WHYBURN [February on Xo, has property B, and is such that lim,-.*» Jlgj{x)dx exists and is different from K. But by the lemma, Jlgj{x)dx must approach the same limit as J h adi{x)dx. It follows from this that lining f b a<f> n {x)dx exists. COROLLARY. If [g n (x)] and [h n {x)] are any two sequences of simple functions that approach f{x) on X 0 and that have property B, then lining Jlg n {x)dx and lining f h ah n {x)dx exist and these limits are equal. THEOREM 6. A necessary and sufficient condition that a function f(x) be summable {in the sense of Lebesgué) is that there exist a sequence [</> n (#)] of simple f unctions which approaches f {x) on X 0 and which has property B. Proof of sufficiency. This may be proved directly or we may obtain it as a consequence of a theorem due to de la Vallée Poussin.* We note that the absolute continuity of the integrals J(f>n{x)dx 1 {n 1, 2, ), is uniform, f since for a given e >0 we may choose ô = e/[2ikf e / 2 ], and S is independent of n. It follows then from de la Vallée Poussin's theorem that f{x) is summable onx. Proof of necessity. Let f{x) be summable on X and let X be subdivided into n equal parts by the points X\y X2y, vv/l 1» Letx 0 =aand x n = b. On IiiXi^x<Xi+i, let<f> n {x) : = fl i^f{x)dx/a il (i = 0,, n l), where Ai = Xi+i~Xi. It followsj from the absolute continuity of ff{x)dx that lim nh>00 4> n {x) =f{x) on X 0. It remains to show that [0n(#)] has property B. Let g{x) = \f{x), h n {x)=f x j+ l g{x)dx/ai on I u (i = 0, 1,,» 1). Let e>0 be assigned and let ô>0 be chosen so that f(e)g{t)dt<e, when e is a measurable subset of X such that w(e)^s. The existence of S follows from the absolute continuity of jtg{i)dt. Let gj{x) =g{x) when \g{x) \^j, gj{x)=j when g{x)>j. The integral J b agj{x)dx is an increasing function of j and has J\g{x)dx as its limit as j becomes infinite. Choose / so that for all j ^ /, J h ag{x)dx fagi{%)dx<e. Let k denote the subset of X on which g{x) >J and let ki denote the subset of k that lies on 7 t -. We have hk)g{x)dx =J2l^of(k i )g{x)dx < e. Now * Transactions of this Society, vol. 16 (1915), p t For definition, see de la Vallée Poussin, loc. cit. p t See W. M. Whyburn, loc. cit., p. 562, for a detailed proof of this.

7 1932.] INTEGRATION OF UNBOUNDED FUNCTIONS 129 *»(*) = Uilt-ki)g(x)dx + f(ki)g(x)dx]/ai ^ J + f(ki)g(x)dx/ai, on /,-. Let M = J+e/8. In order that h n (x) be greater than M e on /» it is necessary that f ( k i )g(x)dx/a i be greater than e/8 and hence A»- be less than [V ]/<fy)go*0^#«from this it follows that the sum I'(n, e) of intervals on which h n (x)>m is of measure less than Y^l=o[a/e]f iki )g(x)dx<eo/es S. Hence fin n,*)g(x)dx == fi, (n, )h n (x)dx<e. Since \<i> n (x)\sh n (x), it follows that the set I(n, e) of intervals on which \4> n (x) \>M is a subset of /'(^> ) and that /j (n, ) 0 n (ff) \dx is less than e for all n. This proves that [0 n (#)] has property B. THEOREM 7. If f{x) is summable on X and if \<t> n (x)\ is any sequence of simple functions which has prpperty B and which approaches f (x) on X 0, then lim nh(00 f h a4>n{x)dx exists and is equal to J b af(x)dx. PROOF. This theorem follows immediately from Theorem 5 and a theorem of de la Vallée Poussin.* A theorem of de la Vallée Poussinf enables us to state Theorem 6 in the following form. THEOREM 8. A necessary and sufficient condition that a function f{x) be summable on X: a^x^b, is that there exist a sequence [<t>n{x) ] of simple functions that approaches f (x) on X 0 and is such that the absolute continuity of the indefinite integrals of these simple f unctions is uniform. We may now state and prove the following theorems which apply when the integrals are taken in senses that are more general than that of Lebesgue. Let ƒ (x) be a function on X : a g x ^ b, and let any definition of the integral of f(x) be given which exists for f(x) and which has the further property that if J(a, b) denotes this integral and c, d, e are any three points of X, then J(c, d), J(c, e), J(e, d) exist and J(c, d) =J(c, e) + J(e, d). * Loc. cit., p. 446, Theorem 2. We note that the proof of Theorem V might be shortened by the use of de la Vallée Poussin's theorems. It seems desirable, however, to give this proof from the Riesz point of view in order that the theorem be available for use in a treatment of Lebesgue integration by the Riesz approach. t Loc. cit., p. 450.

8 130 W. M. WHYBURN [February, THEOREM 9. Let E be an everywhere dense set of points on X and let M denote the subset of X which has the property that if p is any point of M, then lim^p J(p, q)/iq p) exists and equals F(p), where q ranges on the set E. There exists a sequence \<j> n {pc)\ of simple functions such that this sequence converges to F{x) at each point of M and J b a4> n {x)dx = J (a, b) for n = l, 2,. PROOF. Let a method of subdivision of X be chosen in such a way that all of the subdivision points belong to E and the length of the maximum subdivision approaches zero as the number of these subdivisions becomes infinite. Let the points of subdivision, in order of increasing magnitude, for the nth stage of the subdivision be Xo = a, xi, Xi, -, x n = b. On Ii\ Xi^x<x%+u let <t>n(x)=j(xi, Xi+i)/(xi+i Xi), (i = 0, 1,, w 1). Consider [<f> n (x)]>(n=l,2, - ) Let p be any point of M, and for each n let I Pn : p n^x<q n be the subdivision that contains p. Now <l>n(p)=j(pn, qn)/(qn-pn) = [J(pn, P)+AP, ün)]/(qn-pn)=j(pn, P)/iP~Pn) + [J(P, qn)/(ün-p)-j(pn, P) I\P'~P'») ] [(<Z«" P) I'((Z»" Pn)l We note that \{g, n p)/{q n p n )\^\ while its coefl&cient has zero for its limit (since both terms approach F(p)). Since lim^oo J(p n, p)/(p-pn) is F(p), we have lim n^oo 4>niP)-F{p). In case p is a point of subdivision,* we replace the indeterminate ratio in the above formula for <j> n (p) by the limit of this ratio (which is F(p)). Finally, we note that, for each n, fa<l>n(x)dx =Ysl=oJ(Xi, *<+l) = Jfa, &) COROLLARY 1. If E is identical with X, then M is the set of points at which J (a, x) has a derivative with respect to x. In particular, if this derivative is equal to f(x) at a point of M, then [</> n (x) ] converges to f(x) at this point. COROLLARY 2. If fix) is integrable in the sense of Harnack- Lebesgue, Denjoy, or Lebesgue, the conditions of Theorem 9 are satisfied if E is any everywhere dense set on X and M is X 0. Furthermore, Fix) =ƒ(#) on XQ. COROLLARY 3. If fix) is integrable in the D enjoy-khintchine- Young sense and if X can be divided into a countable number of * We define <f> n (b) = 0n(#n-i).

9 i93 2 -] REFLECTIONS IN FUNCTION SPACE 131 measurable sets in such a way that the indefinite integrals (in the D-K-Y sense) are absolutely continuous on each of these* then the conditions of Theorem 9 are satisfied when J (a, b) denotes the D-K-Y integral of ƒ(#), E is any everywhere dense set on X, M is X 0, and F(x) =ƒ(#) on X 0. It is a very simple matter to extend the results of the present paper to cases where the interval of definition X is replaced by any measurable point set E on X. The definition of f(x) is extended to points of X E by letting ƒ (x) be zero at such points. The integral of f(x) on E is then described in terms of the integral of the extended function on X. One could define a simple function on a point set by saying that it is a function which takes on only a finite number of values on this set. This definition is not needed in the present adaptation. THE UNIVERSITY OF CALIFORNIA AT LOS ANGELES REFLECTIONS IN FUNCTION SPACE t BY L. S. KENNISON 1. Introduction. The purpose of this paper is to point out an error,j giving the corrected form of the incorrect theorem referred to below as Delsarte's theorem, also to prove a generalization. However, the contribution made to the geometry of function space may be interesting to some on its own account. We shall consider only functions of one or two variables defined on the interval (a, b) or the corresponding square. All such functions are to be bounded and integrable on the range of definition. We shall denote the continuous arguments on (a, b) by the letters x, s, t, u, written as subscripts or superscripts and shall imply integration on (a, b) with respect to any argument that occurs twice in the same term. * See Khintchine, Comptes Rendus, vol. 152 (1916), p t Presented to the Society, February 28, t See 3 below.